Course website for Michaelmas 2009 Topology and Groups
General information
Course code: C3.1a
Lecturer: Alexander Coward
Teaching assistants: Jessica Banks and Dawid Kielak
Lecture information
Lectures will all be held on Thursdays at 12 noon in L3 (except in week one when we will be in SR2) and on Fridays at 2pm in L2.
University classes will be held at the following times:
- Fridays at 10am in Seminar Room 2
- Fridays at 11am in Room T14 in even weeks and the Higman Room in odd weeks
- Fridays at 4pm at St Catz, Staircase 14 Room 2
Classes will take place in weeks 2 to 8.
Materials
The course will be based on these lecture notes, written by Marc Lackenby. Here are the problem sheets: Sheet 1, Sheet 2, Sheet 3, Sheet 4, Sheet 5, Sheet 6, Sheet 7. Sheet n will be due in at 12 noon on Wednesday in week n+1.
Syllabus
Here is the course synopsis, as specified in the official handbook:
- Homotopic mappings, homotopy equivalence. Simplicial complexes. Simplicial approximation
theorem.
- The fundamental group of a space. The fundamental group of a circle. Application: the
fundamental theorem of algebra. The fundamental groups of spheres.
- Free groups. Existence and uniqueness of reduced representatives of group elements. The fundamental group of a graph.
- Groups defined by generators and relations (with examples). Tietze transformations.
- The free product of two groups. Amalgamated free products.
- The Seifert van Kampen Theorem.
- Cell complexes. The fundamental group of a cell complex (with examples). The realization
of any finitely presented group as the fundamental group of a finite cell complex.
- Covering spaces. Liftings of paths and homotopies. A covering map induces an injection
between fundamental groups. The use of covering spaces to determine fundamental groups:
the circle again, and real projective n-space. The correspondence between covering spaces
and subgroups of the fundamental group. Regular covering spaces and normal subgroups.
Cayley graphs of a group. The relationship between the universal cover of a cell complex,
and the Cayley graph of its fundamental group. The Cayley 2-complex of a group.
- The Nielsen-Schreier Theorem (every subgroup of a finitely generated free group is free) proved using covering spaces.
Everything covered in lectures and classes will be examinable, unless stated otherwise.
Reading
- John Stillwell, Classical Topology and Combinatorial Group Theory (Springer-Verlag,
1993).
Additional Reading
- D. Cohen, Combinatorial Group Theory: A Topological Approach, Student Texts 14
(London Mathematical Society, 1989), Chs 1-7.
- A. Hatcher, Algebraic Topology (CUP, 2001), Ch. 1.
- M. Hall, Jr, The Theory of Groups (Macmillan, 1959), Chs. 1-7, 12, 17 .
- D. L. Johnson, Presentations of Groups, Student Texts 15 (Second Edition, London
Mathematical Society, Cambridge University Press, 1997). Chs. 1-5, 10,13.
- W. Magnus, A. Karrass, and D. Solitar, Combinatorial Group Theory (Dover Publications,
1976). Chs. 1-4.